Steven A. Orszag

Numerical analysis of spectral methods : theory and applications

Spectral Methods Survey of Approximation Theory Review of Convergence Theory Algebraic Stability Spectral Methods Using Fourier Series Applications of Algebraic Stability Analysis Constant Coefficient Hyperbolic Equations Time Differencing Efficient Implementation of Spectral Methods Numerical Results for Hyperbolic Problems Advection-Diffusion Equation Models of Incompressible Fluid Dynamics Miscellaneous Applications of Spectral Methods Survey of Spectral Methods and Applications Properties of Chebyshev and Legendre Polynomial Expansions.

Numerical investigation of transitional and weak turbulent flow past a sphere

This work reports results of numerical simulations of viscous incompressible flow past a sphere. The primary objective is to identify transitions that occur with increasing Reynolds number, as well as their underlying physical mechanisms. The numerical method used is a mixed spectral element/Fourier spectral method developed for applications involving both Cartesian and cylindrical coordinates. In cylindrical coordinates, a formulation, based on special Jacobi-type polynomials, is used close to the axis of symmetry for the efficient treatment of the ‘pole’ problem. Spectral convergence and accuracy of the numerical formulation are verified. Many of the computations reported here were performed on parallel computers. It was found that the first transition of the flow past a sphere is a linear one and leads to a three-dimensional steady flow field with planar symmetry, i.e. it is of the ‘exchange of stability’ type, consistent with experimental observations on falling spheres and linear stability analysis results. The second transition leads to a single-frequency periodic flow with vortex shedding, which maintains the planar symmetry observed at lower Reynolds number. As the Reynolds number increases further, the planar symmetry is lost and the flow reaches a chaotic state. Small scales are first introduced in the flow by Kelvin–Helmholtz instability of the separating cylindrical shear layer; this shear layer instability is present even after the wake is rendered turbulent.

Local energy flux and subgrid-scale statistics in three-dimensional turbulence

Statistical properties of the subgrid-scale stress tensor, the local energy flux and filtered velocity gradients are analysed in numerical simulations of forced three-dimensional homogeneous turbulence. High Reynolds numbers are achieved by using hyperviscous dissipation. It is found that in the inertial range the subgrid-scale stress tensor and the local energy flux allow simple parametrization based on a tensor eddy viscosity. This parametrization underlines the role that negative skewness of filtered velocity gradients plays in the local energy transfer. It is found that the local energy flux only weakly correlates with the locally averaged energy dissipation rate. This fact reflects basic difficulties of large-eddy simulations of turbulence, namely the possibility of predicting the locally averaged energy dissipation rate through inertial-range quantities such as the local energy flux is limited. Statistical properties of subgrid-scale velocity gradients are systematically studied in an attempt to reveal the mechanism of local energy transfer.

Asymptotic Expansion of Integrals

The analysis of differential and difference equations in Chaps. 3 to 5 is pure local analysis; there we predict the behavior of solutions near one point, but we do not incorporate initial-value or boundary-value data at other points. As a result, our predictions of the local behavior usually contain unknown constants. However, when the differential or difference equation is soluble, we can use the boundary and initial data to make parameter-free predictions of local behavior.

Turbulence: Challenges for Theory and Experiment

Research in macroscopic classical physics, such as fluid dynamics or aspects of condensed matter physics, continues to confront baffling challenges that are by no means less demanding than those at the post‐Newtonian frontiers of physics that have been explored since the beginning of this century. This is so even though the basic equations of macroscopic classical physics are known—indeed, have been known for centuries in many cases. Chaos and nonlinear dynamics are examples of the topics that pose new challenges to our understanding of macroscopic classical systems. Turbulence, a phenomenon related to but distinct from chaos, and having strong roots in engineering, has been increasingly in the focus of physics research in recent years.

Spectral Calculations of One-Dimensional Inviscid Compressible Flows

The extension of spectral methods to inviscid compressible flows is considered. Techniques for high resolution treatment of shocks and contact discontinuities are introduced. Model problems that demonstrate resolution of shocks and contact discontinuities over one effective grid interval are given.

Brine fluxes from growing sea ice

It is well known that brine drainage from growing sea ice has a controlling influence on its mechanical, electromagnetic, biological and transport properties, and hence upon the buoyancy forcing and ecology in the polar oceans. When the ice has exceeded a critical thickness the drainage process is dominated by brine channels: liquid conduits extending through the ice. We describe a theoretical model for the drainage process using mushy layer theory which demonstrates that the brine channel spacing is governed by a selection mechanism that maximizes the rate of removal of stored potential energy, and hence the brine flux from the system. The fluid transport through the sea ice and hence the scaling laws for brine fluxes and brine channel spacings are predicted. Importantly, the resulting brine flux scaling is consistent with experimental data for growth from a fixed temperature surface, allowing all parameters in the scaling law to be determined. This provides an experimentally tested first principles derivation of a parameterization for brine fluxes from growing sea ice.

Towards a Renormalized Lattice Boltzmann Equation for Fluid Turbulence

A coarse-grained Lattice Boltzmann equation is examined in which the effects of unresolved (subgrid) scales are formally incorporated within a renormalized relaxation time of the collision operator. Actual values of the renormalized relaxation time are analyzed for the practical case of high-Reynolds flows past slant bodies (airfoils).

Stability of pseudospectral and finite-difference methods for variable coefficient problems

It is shown that pseudospectral approximation to a special class of variable coefficient one-dimensional wave equations is stable and convergent even though the wave speed changes sign within the domain. Computer experiments indicate similar results are valid for more general problems. Similarly, computer results indicate that the leapfrog finite-difference scheme is stable even though the wave speed changes sign within the domain. However, both schemes can be asymptotically unstable in time when a fixed spatial mesh is used.

Resummation techniques in the kinetic-theoretical approach to subgrid turbulence modeling

The potential role of resummation techniques in the kinetic-theory approach to subgrid turbulence modeling is discussed.